Assume x and y are 2 numbers, where xy=24
a- what values could x and y equal if they are whole numbers? Try to list as many as possible.
b- What values could x and y equal if they can be decimals, fractions or whole numbers?
a- Possible values for x and y as whole numbers:
1, 24
2, 12
3, 8
4, 6
6, 4
8, 3
12, 2
24, 1
b- Possible values for x and y as decimals, fractions, or whole numbers:
1, 24
2, 12
3, 8
4, 6
6, 4
8, 3
12, 2
24, 1
-1, -24
-2, -12
-3, -8
-4, -6
-6, -4
-8, -3
-12, -2
-24, -1
These are the possible values assuming the numbers can be any real numbers, including decimals, fractions, or whole numbers.
Dugald substitutes different whole numbers into the expression 5 * (a+a). He notices that the result always ends in the digit 0. Try a few values and explain why thus pattern occurs.
Let's substitute some whole numbers into the expression 5 * (a + a) and observe the results:
For a = 1:
5 * (1 + 1) = 5 * 2 = 10
For a = 2:
5 * (2 + 2) = 5 * 4 = 20
For a = 3:
5 * (3 + 3) = 5 * 6 = 30
For a = 4:
5 * (4 + 4) = 5 * 8 = 40
From these examples, we can see that regardless of the value of a, the result always ends in the digit 0. This pattern occurs because multiplying any number by 5 will always result in a multiple of 5. And when we multiply a multiple of 5 by 2 (which is what we get when we add a + a), the result will always be a multiple of 10, which ends in the digit 0.
In other words, the expression 5 * (a + a) can be simplified as 10 * a. Since a is a whole number, multiplying it by 10 will always yield a result that ends in 0.
a- For whole numbers, the possible values for x and y when xy=24 are:
1 and 24
2 and 12
3 and 8
4 and 6
6 and 4
8 and 3
12 and 2
24 and 1
b- If x and y can be decimals, fractions, or whole numbers, the possible values for xy=24 are:
1 and 24
2 and 12
3 and 8
4 and 6
6 and 4
8 and 3
12 and 2
24 and 1
0.5 and 48
0.6 and 40
1.2 and 20
2.4 and 10
3 and 8
4 and 6
8 and 3
10 and 2.4
20 and 1.2
40 and 0.6
48 and 0.5
To find the possible values of x and y that satisfy the equation xy = 24, we can use a systematic approach.
a) Whole Numbers:
To determine the whole number values of x and y, we need to find all the pairs of factors of 24. Factors are the numbers that divide into 24 evenly without leaving a remainder.
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.
Now, let's list all the possible combinations of these factors that multiply to give 24:
1 × 24 = 24
2 × 12 = 24
3 × 8 = 24
4 × 6 = 24
So, the whole number values of x and y that satisfy xy = 24 are (1, 24), (2, 12), (3, 8), and (4, 6).
b) Decimals, Fractions, and Whole Numbers:
To consider decimal, fraction, and whole number values for x and y, we can use the concept of factors together with the concept of divisors.
The divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.
Now, let's consider all the possible combinations of these divisors as fractions or decimals:
1/24, 2/12, 3/8, 4/6, 6/4, 8/3, 12/2, and 24/1.
0.04167, 0.16667, 0.375, 0.6667, 1.5, 2.67, 6, and 24.
So, the values of x and y can be fractions, decimals, or whole numbers and include the following possibilities: (1, 24), (2, 12), (3, 8), (4, 6), (6, 4), (8, 3), (12, 2), and (24, 1).
Note: These are just a few examples, and there are potentially more combinations depending on the precision required for decimals and the range of fractions considered.